Mandelbrot set in "cross-sections of magnetic field borders" ?

[stereoman]

 
I made this drawing to try to make the concept of gnomon visible.
     The origin is a point in the center of the thalamus , from there , in successive layers , the whole human organism is constructed.
     We can see how the layers have different densities according to function, and we can see how each layer has different forms , also according to their function.
     But this is only a design , in fact all of the layers are interpenetrated into one another , just as in our animal organism contain minerals and  vegetable parts .
      It is this interpenetration which blurs the boundaries in all fields since, from another point of view , being all the same body ,our way of setting limits is entirely artificial , and are never exact.
      Ie, there are marine mammals, and there are also fishes with lungs.
     In a rainbow , we can see different colors, but it is impossible to find the exact boundary between two of them because in reality do not exist.
     So , geometrically, the planet , ends where  the magnetosphere ends, but the planet's magnetosphere is part of the set of magnetospheres in the solar system , forming a unit , etc. .
     I think the Mandelbrot set is like  something ideal and perfect in conditions that can never be given in the real world, in the real world we have the romanescu , we can think of as an ideal fractal, but that may not manifest if not by the power of the gnomon , which always introduces other conditions.
      In other words, the gnomon carries some info that directs the recursive process to a definite shape, as I understand it, the M-set, can´t do this.
       At this point, I think the challenge is to create a digital gnomon wich can be preset and applied  to the M-set or other fractal generator programs.
     ¿Any takers?

[jdebord]

The Zeta function is a new subject for me. Sounds interesting!

Here is an article about the fractal geometry of the zeta function:

http://dhushara.com/DarkHeart/geozeta/zetageo.htm

[kram1032]

Oh, that's nice

Though I wonder why the author thought that this was a good idea. You can't really read the exponent.

Edit: With a function like the Zeta function I'd imagine it's really hard to get proper bailout conditions, right?
Since you can't just say it diverges at some value, for Zeta(+infinity)=1

Or is the bailout condition something like "if Re(Zeta(z))>1, you are definitely inside the set"?

Edit: What happens if you bring the Zeta function's lines onto circles instead? For instance, you could do a Riemann sphere-like transform, mapping the circle |z|=1 to the line Re(z)=1, apply the Zeta function to that, and then do the back-transformation.

[kram1032]

I applied the idea of a "MetaMandelbrotSet" to the Zeta function in Mathematica. This is my result (for now) :

Real Part
Imaginary Part
Absolute

200 iterations, only rendered for the point (0,0)
I'm sure one can do significantly better, though if the full limit Meta Mandelbrotset is a challenge, the correspoinding Zeta Meta must be a nightmare.

[kram1032]

Here is a nice reply on Math.SE concerning the(*) "magnetic Mandelbrot set"
http://math.stackexchange.com/a/140812/49989

Apparently, in those models, x corresponds to the temperature, while c corresponds to how many different spin states there are in a magnetic metal.
In real life, that means that c should be 2 (according to the Ising Model) and x should be somewhere in the range from 250-330K for "real life" materials under every-day-life temperatures.

(*) there isn't "the" magnetic MSet" though - there are many which are all models for different metals, most of which do not actually exist.

So those Minibrots do not actually occur as cross-sections of a magnetic field, but rather as seas of stability within a chaotic function which models the behavior of metals.
And within the real-life-relevant ranges, they do not occur at all.
- We at least never actually had anything that could classify as complex-valued temperature, to my knowledge. And we certainly never had anything with irrational, let alone complex spin. (Electrons, Protons and Neutrons all have +/- 1/2 spin)

Still, perhaps we could come up with some interesting interpretation in which those complex values actually do make sense within real life. As of right now, they seem to mostly be a mathematical curiosity highlighting the complexity surrounding the curie-temperature of magnetic materials.

[Chillheimer]

thanks kram.. although I don't understand all, this answers my initial question, although I had hoped for a 'more wonderful' outcome.
but then again, the fractal nature around is is enough wonder for a lifetime..

[kram1032]

Don't give up hope yet. It wouldn't be the first time that some real value was extended into a complex value in a sensible, physically relevant way

Since temperature is just the average kinetic Energy in a system, and there is a correspondence between energy and pulse (with pulse being a vector quantity while energy is a scalar), perhaps there would be a description in terms of quaternions, where the scalar (real) part would be the average kinetic Energy while the vectorial part would be the average directed pulse. A description like that is possible (if not always ideal) though for a still metal, this average pulse would typically be 0, since if it wasn't, that would mean that the crystal would be moving.

In that case, perhaps (this is purely speculative) the complex model describes moving magnetic materials. If this is correct, maybe the Curie temperature would somehow be dependent on the velocity of the crystal? - Somehow doubtful and I've never actually heard of that but maybe?
It would mean that, for specific, well-defined speeds, you could have magnetic properties in metals that are way too hot to expect such properties.

However, I can't really see how this would apply to the c value: A complex number of spin configurations certainly makes little sense and even an non-integer or negative number is rather hard to grasp. If it just so happens that no islands of stability (regions "inside" the magnetic set) happen along the line c=2, this could still be a valid interpretation.

Or else, perhaps the models that are used are too simple to fully describe the phenomenon (or my naive use of quaternions here really doesn't apply)

[stereoman]

  After reading all that has been said, it strikes me that we haven´t a definition of what is infinity.
   This may be the key issue in my view.
   Numerically, the infinite seems a clear concept, we can keep adding numbers indefinitely without reaching a limit.
   But this has little practical use for us, and also prevents to form a closed concept, the unit, which was the starting point of the ancient mathematicians.
   In ancient mathematics there was not  the zero concept, wich we relate to infinity.
   Geometrically, which is what interests us, because all the numbers have to finish defining forms, a line is the infinite for a point, a plane is the infinite for a line, and a solid is the infinite for a plane , in the same way an hypersolid is infinity for a three-dimensional solid, lets define the hypersolid as the solid extended in time.
    Closely related are logarithmic spirals wich allow for growing while still  inside some " infinite limits", if this can be said
    Seen this way, infinity is a handy and useful concept, my two cents.

[Chillheimer]

  Geometrically, which is what interests us, because all the numbers have to finish defining forms, a line is the infinite for a point, a plane is the infinite for a line, and a solid is the infinite for a plane , in the same way an hypersolid is infinity for a three-dimensional solid, lets define the hypersolid as the solid extended in time.

For me this sequence shows all signs of "fractal behaviour"
I wonder if it itself goes on into infinity..


---the rest is too mathematical for me.. I feel like I don't need to be an expert in everything, so I'll leave that to you guys, instead of answering to stuff I don't understand---

[stereoman]

For me this sequence shows all signs of "fractal behaviour"
I wonder if it itself goes on into infinity..

yours is the right question.
    From one point of view, the sequence continues in an infinite hypothetical, but like with numbers, this does not help us much, things must be closed somewhat to make sense.
     But you will understand If I say that we have the octave, to bring order to  infinite levels of vibration.
     An octave is a closed process, take the color wheel, for us it is a closed circle, but we know that in the natural scale,is not, as there are vibrations above and below of visible colors.,This is also true in music, there are non audible sounds  and vibrations ,some harmonics are beyond our perception.
    Ancient philosophy stated that "man is the measure of everything", this is what we see here, the color wheel can´t be changed in any way, nor can the octave, because they are created by the human senses and mind from whatever is out there.
   In this case, the hypersolid contains all the possibilities for the solid, in the same way, the solid contains all the possibilities for the plane, and the plane contains all the possibilities for the line.
     Imagine how many drawings have been done  and can be done by man trough all times, these are the infinite possibilities the line has, but they all  always remains in the plane.
    

[kram1032]

That's not true: We absolutely have concepts of infinity. In fact, infinitely many of them.
Infinity is highly context dependent and different kinds of infinity occur in different situations.
For instance, the infinity of 1/x as x goes to 0 is "weaker" than the infinity of 1/x²: You need a higher polynomial to "get rid of" the second infinity than for the first. (x²/x²=1 and x/x=1 but x/x²=1/x)
And then there are exponential infinities - ones that cannot be dealt with by any polynomial.

Other classes of infinity are the "infinitely small" or higher infinities which are like infinities of infinities. - Then there are the so-called Surreal Numbers which contain all numerically possible infinities.

It's not so much that we don't have concepts for infinity but rather that most infinities are not very important or too rare to be usually discussed in detail.

The way in which an (n+1)-plane is infinite compared to an n-plane is yet another very different concept of infinity.
We are mathematically also able to talk about infinity-planes - hyper planes of infinite dimensionality.

The most common forms of infinity nowadays have a very well established definition and underlying conception.

[stereoman]

That's not true: We absolutely have concepts of infinity. In fact, infinitely many of them.
Infinity is highly context dependent and different kinds of infinity occur in different situations.
For instance, the infinity of 1/x as x goes to 0 is "weaker" than the infinity of 1/x²: You need a higher polynomial to "get rid of" the second infinity than for the first. (x²/x²=1 and x/x=1 but x/x²=1/x)
And then there are exponential infinities - ones that cannot be dealt with by any polynomial.

Other classes of infinity are the "infinitely small" or higher infinities which are like infinities of infinities. - Then there are the so-called Surreal Numbers which contain all numerically possible infinities.

It's not so much that we don't have concepts for infinity but rather that most infinities are not very important or too rare to be usually discussed in detail.

The way in which an (n+1)-plane is infinite compared to an n-plane is yet another very different concept of infinity.
We are mathematically also able to talk about infinity-planes - hyper planes of infinite dimensionality.

The most common forms of infinity nowadays have a very well established definition and underlying conception.

What is not true?
 Edit. Oh I see, you have definitions of infinite, but this is not the point, the point is how to deal with infinite in geometric terms, I suppose that having so many infinites in your box, there´s room for another one

[kram1032]

Various cases of geometric infinity can also be dealt with.

[stereoman]

Various cases of geometric infinity can also be dealt with.

No doubt, but I think the main problem with the gnomon concept, comes from two different concepts of infinity.
To me, infinity can be and is limited geometrically, but numbers can´t be limited this way, since I must talk with matemathics, and they have it´s own ideas , I try to make mine understandable, in search for the simplest approach.

[youhn]

When the idea of infinity is understood, some things in geometry and math become easier. For example an infinite line is easier to define and understand. Since it has no beginning or end, you don't have to think about it.

Everything in math and geometry is an approximated or extrapolated model from our percepted reality.

When talking about the infinity of numbers ... things get very confusing in my head. Take for example the number pi, which is said to be infinite. This does not refer to it's value, since pi is actually pretty defined. But to exactly express it in our decimal number system... that is a problem. I can agree to the statement that numbers are not infinite. But sets of numbers can be.

Infinity is something we though up to deal with the very big and ungrabbable. It is as real as the concept of "borders".